A Zygmund-type integral inequality for polynomials

  • Abdullah MirEmail author
Open Access


Let P(z) be a polynomial of degree n which does not vanish in \(|z|<1\). Then it was proved by Hans and Lal (Anal Math 40:105–115, 2014) that
$$\begin{aligned} \Bigg |z^s P^{(s)}+\beta \dfrac{n_s}{2^s}P(z)\Bigg |\le \dfrac{n_s}{2}\Bigg (\bigg |1+\dfrac{\beta }{2^s}\bigg |+\bigg | \dfrac{\beta }{2^s}\bigg |\Bigg )\underset{|z|=1}{\max }|P(z)|, \end{aligned}$$
for every \(\beta \in \mathbb C\) with \(|\beta |\le 1,1\le s\le n\) and \(|z|=1.\)
The \(L^{\gamma }\) analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 2016) who under the same hypothesis proved
$$\begin{aligned}&\Bigg \{\int _0^{2\pi }\Big |e^{is\theta }P^{(s)}(e^{i\theta })+\beta \dfrac{n_s}{2^s}P(e^{i\theta })\Big |^ {\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }\\&\quad \le n_s\Bigg \{\int _0^{2\pi }\Big |\Big (1+\dfrac{\beta }{2^s}\Big )e^{i\alpha }+\dfrac{\beta }{2^s}\Big |^{\gamma } \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }\dfrac{\Bigg \{\int _0^{2\pi }\big |P(e^{i\theta })\big |^{\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }}{\Bigg \{\int _{0}^{2\pi }\big |1+e^{i\alpha }\big |^\gamma \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }}, \end{aligned}$$
where \(n_s=n(n-1)\ldots (n-s+1)\) and \(0\le \gamma <\infty \).

In this paper, we generalize this and some other related results.

Mathematics Subject Classification

30A10 30C10 30C15 



The author is extremely grateful to the anonymous referees for many valuable suggestions.


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© The Author(s) 2019

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KashmirSrinagarIndia

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